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If m and n are real numbers and m > n, if m2 + n2, m2 – n2 and 2 mn are the sides of the triangle, then prove that the triangle is right-angled. (Use the converse of the Pythagoras theorem). Find out two Pythagorian triplets using convenient values of m and n.
Concept: Pythagorean Triplet
AB, BC and AC are three sides of a right-angled triangle having lengths 6 cm, 8 cm and 10 cm, respectively. To verify the Pythagoras theorem for this triangle, fill in the boxes:
ΔABC is a right-angled triangle and ∠ABC = 90°.
So, by the Pythagoras theorem,
`square` + `square` = `square`
Substituting 6 cm for AB and 8 cm for BC in L.H.S.
`square` + `square` = `square` + `square`
= `square` + `square`
= `square`
Substituting 10 cm for AC in R.H.S.
`square` = `square`
= `square`
Since, L.H.S. = R.H.S.
Hence, the Pythagoras theorem is verified.
Concept: Pythagoras Theorem
In the given figure, triangle PQR is right-angled at Q. S is the mid-point of side QR. Prove that QR2 = 4(PS2 – PQ2).
Concept: Converse of Pythagoras Theorem
In a ΔABC, ∠CAB is an obtuse angle. P is the circumcentre of ∆ABC. Prove that ∠CAB – ∠PBC = 90°.
Concept: Application of Pythagoras Theorem in Acute Angle and Obtuse Angle
There is a ladder of length 32 m which rests on a pole. If the height of pole is 18 m, determine the distance between the foot of ladder and the pole.
Concept: Pythagoras Theorem
In the figure, ΔPQR is right angled at Q, seg QS ⊥ seg PR. Find x, y.
Concept: Theorem of Geometric Mean
In the given figure, triangle ABC is a right-angled at B. D is the mid-point of side BC. Prove that AC2 = 4AD2 – 3AB2.
Concept: Apollonius Theorem
In an isosceles triangle PQR, the length of equal sides PQ and PR is 13 cm and base QR is 10 cm. Find the length of perpendicular bisector drawn from vertex P to side QR.
Concept: Right-angled Triangles and Pythagoras Property
In the adjoining figure, a tangent is drawn to a circle of radius 4 cm and centre C, at the point S. Find the length of the tangent ST, if CT = 10 cm.
Concept: Right-angled Triangles and Pythagoras Property
In a right angled triangle, right-angled at B, lengths of sides AB and AC are 5 cm and 13 cm, respectively. What will be the length of side BC?
Concept: Converse of Pythagoras Theorem
In an equilateral triangle PQR, prove that PS2 = 3(QS)2.
Concept: Right-angled Triangles and Pythagoras Property
A person starts his trip from home. He moves 24 km in south direction and then starts moving towards east. He travels 7 km in that direction and finally reaches his destination. How far is the destination from his home?
Concept: Pythagoras Theorem
In ∆RST, ∠S = 90°, ∠T = 30°, RT = 12 cm, then find RS.
Concept: Property of 30°- 60°- 90° Triangle Theorem
In the following figure, m(arc PMQ) = 130o, find ∠PQS.
Concept: Angle Subtended by the Arc to the Point on the Circle
In the following figure, secants containing chords RS and PQ of a circle intersects each other in point A in the exterior of a circle if m(arc PCR) = 26°, m(arc QDS) = 48°, then find:
(i) m∠PQR
(ii) m∠SPQ
(iii) m∠RAQ
Concept: Angle Subtended by the Arc to the Point on the Circle
In the given figure, altitudes YZ and XT of ∆WXY intersect at P. Prove that,
- `square`WZPT is cyclic.
- Points X, Z, T, Y are concyclic.
Concept: Angle Subtended by the Arc to the Centre
In the given figure, chord MN and chord RS intersect at point D.
(1) If RD = 15, DS = 4, MD = 8 find DN
(2) If RS = 18, MD = 9, DN = 8 find DS
Concept: Intersecting Chords and Tangents
In the figure Q is the contact point. If
PQ = 12, PR = 8, then PS = ?
Concept: Tangent and Secant Properties
In the adjoining figure, point O is the centre of the cirlcle, seg OM ⊥ chord AB. If OM = 8cm, AB = 12 cm, then find OB.
Concept: Angle Subtended by the Arc to the Point on the Circle
In the adjoining figure chord EF || chord GH.
Prove that chord EG ≅ chord FH.
Fill in the boxes and write the complete proof.
Concept: Inscribed Angle Theorem
